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AP Calculus BC
Unit 4: Contextual Applications of Differentiation
4.1 Interpreting the Meaning of the Derivative in Context
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What does the derivative of a function measure?
Instantaneous rate of change
The notation
f
′
(
x
)
f'(x)
f
′
(
x
)
represents the derivative of
f
(
x
)
f(x)
f
(
x
)
with respect to
x
x
x
.
The instantaneous rate of change of a function at a point is the value of its
derivative
What is the formula for the derivative of a function
f
(
x
)
f(x)
f
(
x
)
?
f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}</latex>
One application of the derivative is to find the instantaneous
velocity
of an object.
The derivative can be used to solve problems involving
rates of change
in contextual applications.
Match the derivative notation with its meaning:
f
′
(
x
)
f'(x)
f
′
(
x
)
↔️ Rate of change of
f
(
x
)
f(x)
f
(
x
)
with respect to
x
x
x
d
y
d
x
\frac{dy}{dx}
d
x
d
y
↔️ Rate of change of
y
y
y
with respect to
x
x
x
D
x
y
D_{x} y
D
x
y
↔️ Derivative of
y
y
y
with respect to
x
x
x
The Leibniz notation for the derivative is
\frac{dy}{dx}
The Leibniz notation is useful when applying the
chain rule
.
What does the operator notation
D
x
y
D_{x} y
D
x
y
mean?
Derivative of
y
y
y
with respect to
x
x
x
The instantaneous rate of change at a point is the value of the
derivative
at that point.
Steps to find the average rate of change of a function
f
(
x
)
f(x)
f
(
x
)
over the interval
[
a
,
b
]
[a, b]
[
a
,
b
]
1️⃣ Calculate
f
(
b
)
f(b)
f
(
b
)
2️⃣ Calculate
f
(
a
)
f(a)
f
(
a
)
3️⃣ Compute
f
(
b
)
−
f
(
a
)
f(b) - f(a)
f
(
b
)
−
f
(
a
)
4️⃣ Divide by
b
−
a
b - a
b
−
a
What is the definition of the derivative as a limit?
f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}</latex>
The derivative of
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
x^{2}
x
2
is 2x
Match the derivative notation with its example:
f
′
(
x
)
f'(x)
f
′
(
x
)
↔️ If
f
(
x
)
=
f(x) =
f
(
x
)
=
x
3
x^{3}
x
3
,
f
′
(
x
)
=
f'(x) =
f
′
(
x
)
=
3
x
2
3x^{2}
3
x
2
d
y
d
x
\frac{dy}{dx}
d
x
d
y
↔️ If
y
=
y =
y
=
e
x
e^{x}
e
x
,
d
y
d
x
=
\frac{dy}{dx} =
d
x
d
y
=
e
x
e^{x}
e
x
D
x
y
D_{x} y
D
x
y
↔️ If
y
=
y =
y
=
sin
(
x
)
\sin(x)
sin
(
x
)
,
D
x
y
=
D_{x} y =
D
x
y
=
cos
(
x
)
\cos(x)
cos
(
x
)
The prime notation
f
′
(
x
)
f'(x)
f
′
(
x
)
represents the rate of change of
f
(
x
)
f(x)
f
(
x
)
with respect to
x
x
x
.
The Leibniz notation
d
y
d
x
\frac{dy}{dx}
d
x
d
y
measures the rate of change of
y
y
y
with respect to x.
What is the instantaneous rate of change of
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
x^{2}
x
2
at
x
=
x =
x
=
3
3
3
?
f
′
(
3
)
=
f'(3) =
f
′
(
3
)
=
6
6
6
What is the notation for the prime notation of a derivative?
f
′
(
x
)
f'(x)
f
′
(
x
)
The prime notation
f
′
(
x
)
f'(x)
f
′
(
x
)
represents the rate of change of f(x)</latex> with respect to x
Prime notation is commonly used when discussing general properties of
functions
or graphing.
What is the notation for the Leibniz notation of a derivative?
\frac{dy}{dx}</latex>
The Leibniz notation
d
y
d
x
\frac{dy}{dx}
d
x
d
y
represents the rate of change of
y
y
y
with respect to x
What is the notation for the operator notation of a derivative?
D
x
y
D_{x} y
D
x
y
The operator notation represents the derivative of
y
y
y
or
f
(
x
)
f(x)
f
(
x
)
with respect to
x
x
x
.
The prime notation
f
′
(
x
)
f'(x)
f
′
(
x
)
represents the rate of change of
f
(
x
)
f(x)
f
(
x
)
with respect to x
What does the Leibniz notation
d
y
d
x
\frac{dy}{dx}
d
x
d
y
represent?
Rate of change of
y
y
y
with respect to
x
x
x
The operator notation
D
x
y
D_{x} y
D
x
y
represents the derivative of
y
y
y
with respect to x
The instantaneous rate of change is defined as the limit of the difference quotient as
h
h
h
approaches 0.
What is the formula for the instantaneous rate of change at
x
=
x =
x
=
a
a
a
?
f
′
(
a
)
=
f'(a) =
f
′
(
a
)
=
lim
h
→
0
f
(
a
+
h
)
−
f
(
a
)
h
\lim_{h \to 0} \frac{f(a + h) - f(a)}{h}
lim
h
→
0
h
f
(
a
+
h
)
−
f
(
a
)
How is the average rate of change calculated over the interval
[
a
,
b
]
[a, b]
[
a
,
b
]
?
\frac{f(b) - f(a)}{b - a}</latex>
The instantaneous rate of change measures how rapidly
f
(
x
)
f(x)
f
(
x
)
is changing at a specific point
The average rate of change is the change in
function values
over an interval.
What is the formula for the power rule of differentiation?
f'(x) = nx^{n - 1}</latex>
What is the formula for the difference rule of differentiation?
f
′
(
x
)
=
f'(x) =
f
′
(
x
)
=
u
′
(
x
)
−
v
′
(
x
)
u'(x) - v'(x)
u
′
(
x
)
−
v
′
(
x
)
The sum rule is used to differentiate the sum of two functions by differentiating each
term
separately.
The sum rule allows us to
differentiate
the sum of two functions term by term.
The difference rule states that if
f
(
x
)
=
f(x) =
f
(
x
)
=
u
(
x
)
−
v
(
x
)
u(x) - v(x)
u
(
x
)
−
v
(
x
)
, then
f
′
(
x
)
=
f'(x) =
f
′
(
x
)
=
u
′
(
x
)
−
v
′
(
x
)
u'(x) - v'(x)
u
′
(
x
)
−
v
′
(
x
)
. If
f
(
x
)
=
f(x) =
f
(
x
)
=
x
4
−
2
x
2
x^{4} - 2x^{2}
x
4
−
2
x
2
, then f'(x) = 4x^{3} - 4x
In economics, what does the derivative of a profit function
P
(
x
)
P(x)
P
(
x
)
represent?
Marginal profit per unit
In physics, the derivative of a position function
s
(
t
)
s(t)
s
(
t
)
gives the velocity at time
t
t
t
.
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